Elastic nematics

Now we return to liquid crystal networks. In addition to the above three postulates, we further assume (4) The total free energy of the network is comprised of elastic, nematic and external stress contributions, i.e.,... 

For instance, for T(p = 10 dyn,ii = 2 x 10 " cm,Fs = 300 statC/cm (1 mC/m ), the threshold field is 0.1 statV/cm, i.e. 3 kV/m. Due to a high value of the Frederiks type distortion in SmC can be observed at extremely low voltage across the cell Uc = dE 30 mV for 10 pm thick cell). However, independently of the field magnitude, after switching the field off, the distortion relaxes to the initial uniform structure, cp(x) = 0. The relaxation time of the distorted structure is owed to pure elastic, nematic-like torque and for small distortion only fundamental Fourier harmonic is important. 

If we disregard the elastic, nematic-like term we would see that the distortion has a threshold character with the threshold field ,/, = 2W/Po. It is easy to understand the AF-F threshold is achieved when the field energy is sufficient for the director in even layers to overcome potential barrier W and change its azimuth from % to 0. Above the threshold, > fa, the uniform ferroelectric structure is installed. 

Allen M P, Warren M A, Wilson M R, Sauron A and Wiliam S 1996 Molecular dynamics calculation of elastic constants in Gay-Berne nematic liquid crystals J. Chem. Phys. 105 2850-8... 

Figure C2.2.11. (a) Splay, (b) twist and (c) bend defonnations in a nematic liquid crystal. The director is indicated by a dot, when nonnal to the page. The corresponding Frank elastic constants are indicated (equation(C2.2.9)).

Undoubtedly the most successful model of the nematic-smectic A phase transition is the Landau-de Gennes model [201. It is applied in the case of a second-order phase transition by combining a Landau expansion for the free energy in tenns of an order parameter for smectic layering with the elastic energy of the nematic phase [20]. It is first convenient to introduce an order parameter for the smectic stmcture, which allows both for the layer periodicity (at the first hannonic level, cf equation (C2.2A)) and the fluctuations of layer position ur [20] ... 

C and I account for gradients of the smectic order parameter the fifth tenn also allows for director fluctuations, n. The tenn is the elastic free-energy density of the nematic phase, given by equation (02.2.9). In the smectic... 

Stannarius R 1998 Elastic properties of nematic liquid crystals 1998 Handbook of Liquid Crystals Vol 2A. Low Molecular Weight Liquid Crystals led D Demus, J Goodby, G W Gray, Fl-W Speiss and V Vill (New York Wiley-VCH)... 

AH distortions of the nematic phase may be decomposed into three basic curvatures of the director, as depicted in Figure 6. Liquid crystals are unusual fluids in that such elastic curvatures may be sustained. Molecules of a tme Hquid would immediately reorient to flow out of an imposed mechanical shear. The force constants characterizing these distortions are very weak, making the material exceedingly sensitive and easy to perturb. 

One prominent example of rods with a soft interaction is Gay-Berne particles. Recently, elastic properties were calculated [89,90]. Using the classical Car-Parrinello scheme, the interactions between charged rods have been considered [91]. Concerning phase transitions, the sohd-fluid equihbria for hard dumbbells that interact additionally with a quadrupolar force was considered [92], as was the nematic-isotropic transition in a fluid of dipolar hard spherocylinders [93]. The influence of an additional attraction on the phase behavior of hard spherocylinders was considered by Bolhuis et al. [94]. The gelation transition typical for clays was found in a system of infinitely thin disks carrying point quadrupoles [95,96]. In confined hquid-crystalline films tilted molecular layers form near each wall [97]. Chakrabarti has found simulation evidence of critical behavior of the isotropic-nematic phase transition in a porous medium [98]. 

The factors Kn are elastic constants for the nematic phase and Icb is the Boltzmann constant. Therefore a combination of molecular electronic structure, orientational order and continuum elasticity are all involved in determining the flexoelectric polarisation. Polarisation can also be produced in the presence of an average gradient in the density of quadrupoles. This is... 

In simple single-site liquid crystal models, such as hard-ellipsoids or the Gay-Berne potential, a number of elegant techniques have been devised to calculate key bulk properties which are useful for display applications. These include elastic constants for nematic systems [87, 88]. However, these techniques are dependent on large systems and long runs, and (at the present time) limitations in computer time prevent the extension of these methods to fully atomistic models. 

Many other interesting examples of spontaneous reflection symmetry breaking in macroscopic domains, driven by boundary conditions, have been described in LC systems. For example, it is well known that in polymer disperse LCs, where the LC sample is confined in small spherical droplets, chiral director structures are often observed, driven by minimization of surface and bulk elastic free energies.24 We have reported chiral domain structures, and indeed chiral electro-optic behavior, in cylindrical nematic domains surrounded by isotropic liquid (the molecules were achiral).25... 

Note 3 The names of Oseen, Zocher, and Frank are associated with the development of the theory for the elastic behaviour of nematics and so the elastic constants may also be described as the Oseen-Zocher-Frank constants, although the term Frank constants is frequently used. 

Elastic deformation of the director, induced by a magnetic induction or electric field, in a uniformly aligned, thin sample of a nematic confined between two surfaces. 

Physical properties of liquid crystals are generally anisotropic (see, for example, du Jeu, 1980). The anisotropic physical properties that are relevant to display devices are refractive index, dielectric permittivity and orientational elasticity (Raynes, 1983). A nematic LC has two principal refractive indices, Un and measured parallel and perpendicular to the nematic director respectively. The birefringence An = ny — rij is positive, typically around 0.25. The anisotropy in the dielectric permittivity which is given by As = II — Sj is the driving force for most electrooptic effects in LCs. The electric contribution to the free energy contains a term that depends on the angle between the director n and the electric field E and is given by... 

For a nematic LC, the preferred orientation is one in which the director is parallel everywhere. Other orientations have a free-energy distribution that depends on the elastic constants, K /. The orientational elastic constants K, K22 and K33 determine respectively splay, twist and bend deformations. Values of elastic constants in LCs are around 10 N so that free-energy difference between different orientations is of the order of 5 x 10 J m the same order of magnitude as surface energy. A thin layer of LC sandwiched between two aligned surfaces therefore adopts an orientation determined by the surfaces. This fact forms the basis of most electrooptical effects in LCs. Display devices based on LCs are discussed in Chapter 7. 

Here a is the elastic stress which arises from the change in the (dynamic) free energy in the macroscopic flow, while o(V) and a(S) are the viscous stresses produced by the polymer-solvent friction and the solvent-solvent friction, respectively. In concentrated isotropic polymer solutions, the elastic stress overwhelms the viscous stresses, so the latter are often neglected. However, it should be noticed that the viscous stresses may become significant in more dilute solutions as well as in nematic solutions where the elastic stress diminishes. 

Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

It has been shown 65,68) that the threshold voltage is a function of the dielectric anisotropy Ae and the elastic constants of splay (ku), twist (k22) and bend (k33) deformation of the nematic phase (Fig. 17) ... 

Fig. 17a-c. Elastic constants for a splay b twist c bend deformations of a nematic phase. The full lines represent the director... 

We will discuss some preliminary results, which have been performed recently l01). In Fig. 39a the results for polymer No. 2d of Table 10 are shown, which were obtained by torsional vibration experiments. At low temperatures the step in the G (T) curve and the maximum in the G"(T) curve indicate a p-relaxation process at about 120-130 K. Accordingly the glass transition is detected at about 260 K. At 277 K the nematic elastomer becomes isotropic. This phase transformation can be seen only by a very small step in G and G" in the tail of glass transition region, which is shown in more detail in Fig. 39 b. From these measurements we can conclude, that the visco-elastic properties are largely dominated by the properties of the polymer backbone the change of the mesogenic side chains from isotropic to liquid crystalline acts only as a small disturbance and in principle the visco-elastic behavior of the elastomer... 

The theory of nematic liquid crystal deformation, forced by an electric field is well developed and permits to establish the relationship between the threshold voltage U, causing sample orientation, with Ae and elasticity constants of a liquid crystal (Kn). For the main S and B types of deformation the equation is the following27 ... 

U values are substantially higher (U = 20-60 V). This was interpreted to be a result of significant differences in elastic constants of polymeric nematics with varying spacer lengths. 

Theoretical analysis indicates that occurrence of such convective instabilities depends on anisotropy of electrical conductivity and dielectric properties in the initial aligned nematic material. That is, conductivity parallel to the direction of alignment must differ from conductivity perpendicular to this direction. Calculation of the stability condition requires knowledge not only of these anisotropic electrical properties but also of anisotropic elastic and viscous properties which oppose disruption of the alignment and flow. 

Sathyanarayana P, Mathew M, Sastry VSS, Kundu B, Le KV, Takezoe H, Dhara S (2010) Splay bend elasticity of a bent-core nematic liquid crystal. Phys Rev E (Rapid) 81 010702(R)-... 

Kundu B, PratibhaR, MadhusudanaNV (2007) Anomalous temperature dependence of elastic constants in the nematic phase of binary mixtures made of rodlike and bent-core molecules. Phys Rev Lett 99 247802-1-4... 

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